Strong-Field QED Workshop 2024

• The workshop is a premier gathering of experts focused on nonperturbative SFQED, combining theoretical advances with practical experimental methodologies using high-intensity lasers.
• It highlights state-of-the-art computational techniques, including LCFA, LMA, and spin-resolved QED-PIC simulations, to accurately model multiphoton processes and electron–photon dynamics.
• The event provides actionable guidance to identify quantum signatures such as nonlinear Compton edges, effective mass shifts, and pair production thresholds in extreme electromagnetic fields.

The Strong-Field QED Workshop 2024, held at DESY, assembled leading experts in theoretical and experimental quantum electrodynamics in extreme electromagnetic fields. The event focused on nonperturbative quantum field processes that emerge when the intensity parameter $a_0$ and the quantum nonlinearity parameter $\chi$ approach or exceed unity, causing perturbative QED to break down and new high-field dynamics to manifest. Special emphasis was placed on practical guidance for experimentalists, emerging computational techniques, and the translation of advanced SFQED theory—especially the use of Volkov states and the locally-constant-field (LCFA)/locally-monochromatic (LMA) approximations—into robust experimental signatures.

1. Physical Regimes and Key Parameters in Strong-Field QED

Strong-field quantum electrodynamics (SFQED) governs the behavior of electrons, positrons, and photons in background fields near or beyond the Schwinger critical field, $$E_{\mathrm{cr}} = m^2c^3/(e\hbar) \approx 1.32 \times 10^{18}\,\mathrm{V/m}$$ (Kropf et al., 13 Nov 2025, Sarri et al., 3 Apr 2025). Characteristic nonlinearity parameters include

• Intensity parameter $a_0 = eE/(m\omega_L c)$ (sometimes called $\xi$), quantifying the work of the laser over a Compton wavelength in units of an optical photon (Kropf et al., 13 Nov 2025, Pentia et al., 2023);
• Quantum parameter $$\chi = (e\hbar/m^3c^4)\sqrt{-(F_{\mu\nu}p^\nu)^2} \approx \gamma E_\perp / E_{\rm cr}$$ for electrons or $$\chi_\gamma = (\hbar\omega_\gamma/mc^2) E_\perp/E_{\rm cr}$$ for photons.

In the regime $a_0 \gtrsim 1$, electron motion becomes relativistic in the transverse field, leading to significant absorption of many background photons per quantum event. For $\chi \gtrsim 1$, quantum recoil and radiation reaction drive fundamentally stochastic, nonperturbative dynamical effects, and higher-order radiative corrections $\sim \alpha \chi^{2/3}$ proliferate.

The Furry picture, wherein the classical electromagnetic background is treated exactly (e.g., via Volkov states for electrons in a plane wave), forms the foundation for all modern SFQED predictions (Kropf et al., 13 Nov 2025, Sarri et al., 3 Apr 2025).

2. Dominant Processes and Theoretical Tools

Strong-field QED phenomena are dominated by multiphoton processes:

• Nonlinear Compton scattering: $e^- + n\gamma_L \rightarrow e^- + \gamma'$—the emission of high-energy photons resulting from absorption of many laser photons, described by the Volkov solution and characterized by harmonics in the emitted spectrum (Pentia et al., 2023, Kropf et al., 13 Nov 2025).
• Nonlinear Breit–Wheeler pair production: $\gamma + n\gamma_L \rightarrow e^+ + e^-$—pair creation via multiple background photons, with thresholds and rates depending nontrivially on $a_0$, $\chi_\gamma$ (Sarri et al., 3 Apr 2025).
• Trident processes and vacuum polarization: Higher order in $\alpha$, but opening new channels in ultra-strong fields (Kropf et al., 13 Nov 2025).

Process rates in the regime $a_0 \gg 1$ and $\chi \gg 1$ are most economically handled using the LCFA and LMA. In LCFA, one approximates the field locally as a constant crossed field, leading to closed-form rates containing modified Bessel functions $K_{2/3}$ and $K_{1/3}$. The LMA extends this for nearly monochromatic backgrounds, introducing finite-bandwidth corrections (LMA$^+$) that significantly improve agreement with exact S-matrix calculations for finite pulses (Larin et al., 5 Jun 2025).

A central, nontrivial feature is the so-called "mass shift"—often interpreted as an effective electron mass $m_\text{eff} = m\sqrt{1 + z_f}$, where $z_f = 2U_p/(mc^2)$ and $U_p$ is the ponderomotive potential. However, rigorous analysis reveals this is not a true change in rest mass but instead accounts for the four-potential associated with the electron's required potential energy and momentum to remain in the plane wave. When the ponderomotive four-potential is included covariantly, kinematics remain conventional, and no intrinsic mass shift remains (Reiss, 2013).

Refractive processes with external photons (e.g., vacuum birefringence) in strong backgrounds also exhibit a quasi-conservative property: the net number of background photons exchanged is $O(1)$, even for ultra-relativistic intensities. This is rooted in the short formation length (uncertainty-limited) over which the virtual $e^+e^-$ pairs propagate and precludes large-scale depletion or multi-photon absorption (Piazza, 2013).

3. Experimental Methodologies and Signatures

The workshop emphasized methodologies for robust experimental discrimination between perturbative and nonperturbative processes:

Setup/Technique	Key Parameter Regime	Signature/Observable
High-energy electron + Petawatt laser (Turner et al., 2022, Olofsson et al., 2023, Sarri et al., 3 Apr 2025)	$a_0 \sim 10$–100, $\chi_e \sim 0.1$–10	Nonlinear Compton edge, harmonics, stochastic recoil, onset of pair production
All-optical (LWFA + ultraintense laser) (Pentia et al., 2023)	$a_0 \sim 50$, $\chi_e \sim 0.1$–1	Transition from classical to quantum radiation, GeV positrons
Ultra-relativistic beam–plasma or beam–beam (Matheron et al., 2022, Zhang et al., 2023)	$\chi \sim 10$–100	Quantum beamstrahlung peak, multi-GeV $\gamma$-ray emission, pair cascades

Key experimental observables include shifted Compton edges (intensity-induced mass shift/harmonic cutoff), upscaled pair yields (tunneling vs multiphoton scaling), polarization rotation/birefringence in probe beams, and spectral shape changes due to stochastic recoils.

A major technical challenge is disentangling shot-to-shot field and beam fluctuations from genuine quantum effects. Approximate Bayesian Computation (ABC) schemes have been demonstrated to robustly extract nonperturbative parameters (e.g., effective mass shift $\theta$ in $$m_\mathrm{eff}^2 = m^2[1+\theta \alpha_f \chi^{2/3}]$$) even with imperfect knowledge of collision parameters (Olofsson et al., 2023). Shot-by-shot moment analysis of photon spectra can unambiguously separate mass-shift signatures from experimental noise.

SFQED relevance to future colliders is manifest in beamstrahlung: in the low-disruption, high-$\chi$ regime, the analytic spectrum shows a high-energy peak close to the beam energy and is highly sensitive to quantum corrections (cascade onset, pair depletion) (Zhang et al., 2023). The same is true for beam–plasma strategies that reach similarly high $\chi$ using a single bunch (Matheron et al., 2022).

4. Numerical Simulation and Computational Advances

As SFQED processes are highly stochastic and couple intricate QED rates with kinetic plasma evolutions, scalable and reliable simulation tools are required. Two main approaches were covered:

• Optimized Event Generators: By minimizing rate evaluations per QED event (using the independence of the exponential waiting-time distribution), and tabulating polynomial/rational approximations for emission and pair rates (LCFA), codes such as hi-$\chi$ and SFQEDtoolkit enable an order-of-magnitude speedup over standard rejection-sampling even at high $\chi$ (Panova et al., 25 Sep 2024, Volokitin et al., 2023, Montefiori et al., 2023). OpenMP/MPI parallelization and vectorization strategies achieve near-ideal scaling on modern HPC clusters.
• Spin and Polarization-Resolved QED-PIC: Full inclusion of spin-polarized, angle-resolved strong-field processes in PIC algorithms enables prediction of polarization transfer, far-from-equilibrium cascades, and nonlinear quantum feedback in plasma (Wan et al., 2023). BMT equations for spin, coupled to radiative event samplers, are embedded in time-stepping loops to track all relevant degrees of freedom, with validation against single-particle QED codes.
• Lattice-Symplectic Algorithms: Gauge-invariant, symplectic semi-discrete field-theory solvers have been constructed for secular accurate, ab initio real-time simulation of quantized Dirac–Maxwell (SFQED) fields, supporting Schwinger pair production and vacuum Kerr effect (Chen et al., 2019). The geometric discretization ensures energy/momentum conservation and exact gauge invariance.

Systematic validation shows that modern kernels can recover analytic QED predictions (photon and pair yields, energy spectra) to $<0.5\%$ error over experimental timescales.

5. Experimental and Community Programmes

The workshop placed SFQED in the context of major experimental campaigns and next-generation particle accelerator design (Sarri et al., 3 Apr 2025):

• Facilities such as ELI-NP, BELLA PW, and Apollon deliver intensities of $I\sim 10^{22}\!-\!10^{24}$ W/cm$^2$ and pulse durations $<50$ fs. Electron beams with $\gamma_e \sim 10^3-10^5$ provide access to $\chi \gtrsim 1$.
• The European Roadmap for Particle Physics identifies SFQED as a discovery channel for phenomena like vacuum birefringence, nonlinear Breit–Wheeler pair production, and potential breakdown of perturbative QED at $\alpha\,\chi^{2/3}\sim 1$.
• Detector requirements include large dynamic range, sub-micron overlap and femtosecond synchronization, and polarimetric sensitivity for birefringence.
• High-statistics, systematic parametric scans are essential for benchmarking both analytic theory and large-scale QED-PIC simulations.

There is direct relevance to beamstrahlung and quantum radiation reaction in multi-TeV lepton collider designs, where control of high-$\chi$ processes will be critical for luminosity and background management.

6. Open Challenges and Theoretical Outlook

Critical challenges discussed include:

• The limits of the LCFA/LMA approximations: both suffer in ultra-short pulses, at low photon energies (where formation length exceeds the field scale), and in highly structured beams (Larin et al., 5 Jun 2025).
• The physical interpretation and observability of the "mass shift": correctly accounting for ponderomotive energy avoids misinterpretation of experimental signatures (Reiss, 2013).
• Strategies for nonperturbative validation: ABC and advanced data-driven parameter inference will be key for robust experimental tests in the presence of uncontrolled fluctuations (Olofsson et al., 2023).
• Extending simulation: Hybrid quantum–classical, error-mitigated quantum simulation of SFQED Hamiltonians is in early stages but shows promise for future regimes inaccessible to classical computation (Hidalgo et al., 2023).

Longer-term goals include pushing experimental $\chi$ beyond the current limit, observing vacuum pair production (Schwinger effect) in the lab, and direct tests of Ritus–Narozhny breakdown of QED perturbation theory ($\alpha \chi^{2/3} \sim 1$) (Sarri et al., 3 Apr 2025).

7. Summary Table: Regimes, Methods, and Signatures

Regime	Methodology	Key Signature / Observable
$a_0\lesssim 1$	Standard QED, perturbation	Linear Compton; weak pair production
$a_0 \gtrsim 1$, $\chi \ll 1$	Volkov/LCFA, multiphoton S-matrix	Nonlinear Compton edges, effective mass effect in spectrum
$\chi \gtrsim 1$	LCFA, LMA$^+$, kinetic QED-PIC	Stochastic photon emission, onset of pair creation, effective mass not directly observable
$\chi \gg 1$, $a_0\gg 1$	Bayesian data analysis; nonperturbative approximations; ABC	Cascade onset, correlated multiplet spectra, vacuum birefringence, threshold shift suppression

• "Primer of Strong-Field Quantum Electrodynamics for Experimentalists" (Kropf et al., 13 Nov 2025)
• Reiss, "The mass shell of strong-field quantum electrodynamics" (Reiss, 2013)
• Larin, Seipt, "Extended locally monochromatic approximations of Strong-Field QED processes" (Larin et al., 5 Jun 2025)
• Sarri et al., "Input to the European Strategy for Particle Physics: Strong-Field Quantum Electrodynamics" (Sarri et al., 3 Apr 2025)
• Panova et al., "High-Performance Implementation of the Optimized Event Generator for Strong-Field QED Plasma Simulations" (Panova et al., 25 Sep 2024)
• Smeenk/Titi, radiation pressure experiments (Reiss, 2013), and references therein.

These resources and the presented computational/formal advances provide a roadmap for both immediate and future exploration of QED phenomena at the intensity frontier.